Yohan Cabaye and Loic Remy made the most of the Magpies' control in the first half to open up a commanding lead, and the Hammers were unable to build on the lifeline of a Mike Williamson own goal as they conceded a third late on.
It took just 16 minutes for the visitors to break the deadlock, after three Premier League games without a win or a goal, with Yoan Gouffran allowed to carry the ball towards the box and tee up Cabaye to calmly stroke into the bottom corner.
Remy doubled that advantage on 33 minutes, with a Moussa Sissoko cross picking him out six yards out - and the Frenchman made no mistake as he controlled on his chest and poked under Adrian.
West Ham found a route back into the tie on the stroke of half-time, as Carlton Cole bundled past a sliding Tim Krul and the unfortunate Williamson saw the ball ricochet off his shin and over the line, but it was to be an afternoon of further home discomfort for the Hammers.
Cabaye made sure of that deep into second half stoppage-time as he whipped a 20-yard free-kick over the wall and past a full stretch Adrian, who could only touch the effort onto the inside of the post.
The Magpies in full flow were great to watch but they will face few more accommodating back fours than Razvan Rat, James Collins, Roger Johnson and Matt Taylor.
Cabaye and Remy benefited from some woeful defending and had the Magpies been just a little more clinical, the contest would have been over by half-time.
Injury-hit West Ham's shortcomings in defence were exposed in the opening minutes when Cabaye threaded a terrific pass to Remy.
The Frenchman outfoxed Johnson and Collins and, having created enough room to take a shot, sent the ball narrowly wide of the left post.
When Cabaye was given the opportunity to put Newcastle ahead in the 16th minute, he seized his chance.
Yoan Gouffran darted infield and found Cabaye, who took two touches and passed the ball into the net with West Ham's passive defence all but waving him through.
It was turning into the Cabaye show as the French midfielder set up Moussa Sissoko with a long pass, which should have been buried, before shooting just wide himself.
The half-hour mark saw West Ham's first chance, but Stewart Downing stabbed the ball straight into the arms of Tim Krul.
Any hope that Downing's attempt might inspire the Hammers faded almost immediately as Newcastle capitalised on more shambolic defending to surge 2-0 ahead.
Sissoko floated in a cross that should have been cleared with ease, but instead Remy was able to glide unmarked into the six-yard box and stab home.
Remarkably, given the run of play, West Ham pulled one back deep into first-half injury time and this time Newcastle's defence was at fault.
Rat provided the cross to Cole, who chested the ball down and muscled his way through two defenders, Williamson adjudged to have got the final touch as the ball found its way past Krul.
The goal seemed only to be a momentary lapse in concentration as Cabaye tested Adrian early in the second half, but slowly West Ham began to display some stomach for the fight.
A hint of complacency had crept into Newcastle's play and it was now the Hammers who were driving forward.
Cole should have rounded off Matt Taylor's hard work by dispatching the equaliser but instead he blundered over a simple chance, steering the ball wide with the target at his mercy.
Carroll's first appearance of the season at Upton Park was greeted by a loud cheer, but when the stage beckoned for the England forward to deliver he skied the ball into the stands.
Fittingly, it was the brilliant Cabaye who supplied Newcastle's third goal in the 95th minute with a spectacular free-kick strike from the edge of the box.